, 30 31. 2009.

1. 1. a, b n 22 2nba + . .

22 nba +

2. - . CD ( ABD ), P () CD. AP? 3. n n (n 2) . . . .

4. 2009- 5 8. 2008- 11.

2. 5. AB , AB .0BA 6. {1, 2, 3, , 2009} 1005 , 2009 2010. 1005 . 7. ABCD ,

CDBABDADBCBD == 2,2 AB = CB. ABCD . 8. x, y, z

.21

11

11

11

222=+++++ zyx

.31

21

21

21

333

1. c

a .

( .2 222 Nccnba =+ ) 222 2nbac = ac >,

2

222 acnb = =+=+=+

22

2222222 caacanba

.224

22 222222

+

+=++++ accaaacccaca a c ,

, ca + ac 2

ca + 2

ac . 2. . QAPQ : DPAQPD = , QPD

DPA ADPDQDPQ :: = . , QDA DPA ADPDQADQ :: = .

2

==ADPD

QADQ

QDPQ

QAPQ .

- . ABC a 22aAD = rPD 2= , r . )(2 cbarS ABC ++=

.222

2

+=++=a

aaaar

,

( ) ( )223412412

2

2222

22

2

2

==

+=

+=a

a

QAPQ .

P

C

Q

B

D

3. -. . 2 . 1 n. . . , 2 . , 4 . . 2 2 ( ). , . 4 . 4. 11 11. : - a b, - ,

.

)12( +n

)2( n

babaabaabab KK ababbabbaba KK , ,

. , - . , . , . , , . , - , .

)12( +naa bb

aaa

)2( n

. 3=n , . , . , )12( +n - . , - , . , .

)12( n

5. BAAB 0| ( )( ))100(1010|10 BABABA +++ ,

BBA 9|10 + . , 0=B A ( ).

A10{ 9,7,6,4,3,2,1B } A .

, 4=B , 410 +A 36 A . , 5=B 510 +A 45 1=A 4=A . 8=B , 810 +A 72 1=A . , :

10, 15, 18, 20, 30, 40, 45, 50, 60, 70, 80 90. 6. 1005 . 2009 1005, 2009 2010. 1 2009

S nn

1005 ,1004 ,1006 ,1003 ,1007 , ,3 ,2007 ,2 ,2008 ,1 ,2009 K . 2009 2010. , . S 1005=S , 2009

. , , S 1005 ,1006 , ,2007 ,2008 K S

S}.1005 ,1006 , ,2007 ,2008,2009{ K=S

7. ==ADB , CDB . 2,2 == ABDCBD . ( )BABk , ,

D . BD ( ). APE ,

A

B

C

D

P

2 2

=APE . =CPE . PCD . , BD . D , BD . 2=2 , =

ABCD AB = BC, , .

.90 o== CEBAEB

8. zyx ,, 1.

11

32

21

23 +++ xx ,

( ) ( ) 01312 23 > xx , ( )( ) ( )( ) 0113112 2 >+++ xxxxx ,

0)12)(1( 2 > xxx . , . 01>x 12 222 +>+= xxxx

11

32

21

23 +